On Resolving Singularities

Let V be an irreducible affine algebraic variety over a field k ofcharacteristic zero, and let (f_0,...,f_m) be a sequence of elements of thecoordinate ring. There is probably no elementary condition on the f_i and theirderivatives which determines whether the blowup of V along (f_0,...,f_m) isnonsingular. The result is that there indeed is such an elementary condition,involving the first and second derivatives of the $f_i,$ provided we admitcertain singular blowups, all of which can be resolved by an additional Nashblowup. There is is a particular explicit sequence of ideals R=J_0, J_1, J_2,...\subset R so that V_i=Bl_{J_i}V is the i'th Nash blowup of V, with J_i|J_{i+1}for all i. Applying our earlier paper, V_i is nonsingular if and only if theideal class of J_{i+1} divides some power of the ideal class of J_i. Thepresent paper brings things down to earth considerably: such a divisibility ofideal classes implies that for some N\ge r+2J_i^{N-r-2}J_{i+1}^{r+3}=J_i^NJ_{i+2}. Yet note that this identity in turn implies J_{i+2} is a divisor of somepower of J_{i+1}. Thus although $V_i$ may fail to be nonsingular, when theidentity holds the {\it next} variety V_{i+1} must be nonsingular. Thus theNash question is equivalent to the assertion that the identity above holds forsome sufficiently large i and N.